Gaussian quadrature involving Einstein and Fermi functions with an application to summation of series
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 by Walter Gautschi and Gradimir V. Milovanović PDF
 Math. Comp. 44 (1985), 177190 Request permission
Abstract:
Polynomials ${\pi _k}( \cdot ) = {\pi _k}( \cdot ;d\lambda )$, $k = 0,1,2, \ldots$, are constructed which are orthogonal with respect to the weight distributions $d\lambda (t) = {(t/({e^t}  1))^r}\;dt$ and $d\lambda (t) = {(1/({e^t} + 1))^r}\;dt$, $r = 1,2$, on $(0,\infty )$. Momentrelated methods being inadequate, a discretized Stieltjes procedure is used to generate the coefficients ${\alpha _k},{\beta _k}$ in the recursion formula ${\pi _{k + 1}}(t) = (t  {\alpha _k}){\pi _k}(t)  {\beta _k}{\pi _{k  1}}(t)$, $k = 0,1,2, \ldots$, ${\pi _0}(t) = 1$, ${\pi _{  1}}(t) = 0$. The discretization is effected by the GaussLaguerre and a composite Fejér quadrature rule, respectively. Numerical values of ${\alpha _k},{\beta _k}$, as well as associated error constants, are provided for $0 \leqslant k \leqslant 39$. These allow the construction of Gaussian quadrature formulae, including error terms, with up to 40 points. Examples of npoint formulae, $n = 5(5)40$, are provided in the supplements section at the end of this issue. Such quadrature formulae may prove useful in solid state physics calculations and can also be applied to sum slowly convergent series.References

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Additional Information
 © Copyright 1985 American Mathematical Society
 Journal: Math. Comp. 44 (1985), 177190
 MSC: Primary 65D32; Secondary 33A65, 65A05, 65B10, 8108, 8208
 DOI: https://doi.org/10.1090/S00255718198507710391
 MathSciNet review: 771039